目錄
1 MANIFOLDS
Preliminaries
Differentiable Manifolds
The Second Axiom of Countability
Tangent Vectors and Differentials
Submanifolds, Diffeomorphisms, and the Inverse Function Theorem
Implicit Function Theorems
Vector Fields
Distributions and the Frobenius Theorem
Exercises
2 TENSORS AND DIFFERENTIAL FORMS
Tensor and Exterior Algebras
Tensor Fields and Differential Forms
The Lie Derivative
Differential Ideals
Exercises
3 IE GROUPS
Lie Groups and Their Lie Algebras
Homomorphisms
Lie Subgroups
Coverings
Simply Connected Lie Groups
Exponential Map
Continuous Homomorphisms
Closed Subgroups
The Adjoint Representation
Automorphisms and Derivations of Bilinear Operations and Forms
Homgeneous Manifolds
Exercises
4 INTEGRATION ON MANIFOLDS
Orientation
Integration on Manifolds
de Rham Cohomology
Exercises
5 SHEAVES, COHOMOLGY, AND THE DE RHAM THEOREM
Sheaves and Presheaves
Cochain Complexes
Axiomatic Sheaf Cohomology
The Classical Cohomology Theories
Alexander-Spanier Cohomology
de Rham Cohomology
Singular Cohomology
Cech Cohomology
The de Rham Theorem
Multiplicative Structure
Supports
Exercises
6 THE HODGE THEOREM
The Laplace-Beltrami Operator
The Hodge Theorem
Some Calculus
Elliptic Operators
Reduction to the Periodic Case
Ellipticity of the Laplace-Beltrami Operator
Exercises
BIBLIOGRAPHY
SUPPLEMENT TO BIBLIOGRAPHY
INDEX OF NOTATION
INDEX
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